Numerical Simulation of the Dynamics of Turbulent Swirling Flames

2.2 The Energy Spectrum and Turbulent Length Scales
The kinetic energy (per unit mass) produced by the instantaneous velocity turbulent
fluctuations is defined by:
k ≡ 1 2 u′ i u′ i = 1 2
(
u ′ x 2 + u ′ y 2 + u ′ z
2 ) . (2.5)
The mean value of the instantaneous kinetic energy of the turbulent fluctuations
is called the turbulent kinetic energy [164] and defined by:
〈k〉 ≡ 1 2
〈 〉
u
′
1
(〈 〉 〈 〉 〈 〉)
i u′ i = u ′ 2
x + u ′ 2
y + u ′ 2
z . (2.6)
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2.2 The Energy Spectrum and Turbulent Length Scales
Eddies with various length scales are present in turbulent flows producing
different amounts of kinetic energy. An example to illustrate the variation of
scales in a turbulent flow is shown in Figs. 2.2 and 2.3. Inside the flow, the
eddies are mixed and continually forming and breaking down. In this process,
the largest scale eddies interact with and extract energy from the mean flow
mainly by vortex stretching (due to mean velocity gradients) [201, 208] and
transfer it to the smaller scales. The large eddies break down into smaller ones,
which break down into yet smaller eddies, until they become small enough
that viscous dissipation effects dominate and simply dissipate into internal
energy [130]. This concept, known also as the energy cascade, was introduced
by Richardson in 1922 [170]. He summarized this process with the following
verse:
Big whorls have little whorls,
Which feed on their velocity,
And little whorls have lesser whorls,
And so on to viscosity.
From the concept of energy cascade, the turbulent kinetic energy (Eq. (2.6))
depends on the energy produced by the different eddies. The contribution
of the different scales on the turbulent kinetic energy can be defined by its
spectrum in wave number space E(κ). Then the turbulent kinetic energy is
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